Order Iterative Methods Research Articles

Effect of Equivalence Ratio on Composition and performance of Biogas and Gasoline Exhaust from Spark Ignition Engine by Mathematical Modeling

This paper presents the numerical computationnal of pressure, temperature and exhaust characteristics of spark ignition engine with biogas as fuel. The solution of non-linear combustion equation systems have been computed, that based on a quasi-one-dimensional engine model, high order iteration method with the equilibrium constants method. Computer program was used to calculate the mole fractions of 10 combustion products when biogas and gasoline fuel are burnt along with variable equivalence ratios. In cylinder chamber model is based on the classical two-zone approach, wherein parameters like heat transfer from the cylinder, blow by energy loss and heat release rate are also considered and calculated. Biogas is defined as fuel produced from using anaerobic digestion of biodegradable or waste materials and the constituents are C 5 H 7 O 2 N, CH 4 , CO 2 N 2 H 2 O of biogas and C 7 H1 7 of gosoline. Which general fuel model is specified by way of its C a H b O c N d values. The curve-fitted coefficients of energy were then employed to simulate air and fuels data along with frozen composition and practical chemical equilibrium routines from Gill data. The calculated data were used to plot the various pressure and temperature with the crank angle of each step of four stroke engine cycle and combustion products versus equivalence ratio. All results were compared with gasoline as reference fuel in the spark ignition engine according to the same numerical method.

Ptychographic iterative engine with partially coherent illumination for weakly scattering samples

As an expansion of coherent diffraction imaging, ptychographic iterative engine (PIE) not only inherits advantages such as ultra-high resolution and compact optical system, but also expands the field of view in quantitative imaging, thus PIE is widely used in short wavelength imaging such as X-ray and electron beam imaging, and then extended to visible light field. However, PIE requires coherent illumination for both phase and amplitude retrieval, while traditional X-ray or electron beam sources often cannot satisfy this strict coherent condition, which leads to poor-quality information retrieval with low signal-to-noise ratio. Though several proposed methods such as multiple wavelength and multi-mode algorithms can eliminate incoherency influence to some extent, various details such as quantitative spectrum of illuminating source should be obtained before information retrieval, which complicates computing procedures. In addition, it is hard to acquire the spectrum of the illuminating source in most cases. In order to acquire high-quality information based on PIE with partially coherent illumination, a newly designed enhanced phase retrieval method for weakly scattering samples in PIE with partially coherent illumination is presented in this paper, in which only the bright field of the diffraction patterns is used in the iterative procedures mimicking the coherent cases especially for weakly scattering samples without any prior illuminating details. The bright field area can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. While the dark-field area generated by interference of diffracted beams cannot satisfy the requirement for coherence, therefore, dark-field diffraction patterns should be excluded in sample information extraction and only the bright field is used for phase retrieval via iterative process. Compared with the proposed multiple wavelength and multi-mode algorithms, the proposed method can simplify sample reconstructing procedures due to needing no prior knowledge. Moreover, in order to enhance the information of weakly scattering samples in retrieval, high order iteration method is also introduced in the paper. Since the bright field can be regarded as purely coherent diffraction patterns composed of a strengthened zeroth order beam and a weakened diffracted beam. For weakly scattering sample, the weakened diffracted beam is much lower than zeroth order beam, thus it is difficult to acquire high-contrast information with classical PIE algorithms. Introducing high order iterative tactic, the contrast of weakly scattering sample is obviously improved and the details of weakly scattering sample can be retrieved clearly. Both theoretical analysis and numerical simulations are illustrated in detail, proving the robustness and availability of the designed method: high-contrast phase information can be obtained with the proposed method, while traditional phase retrieval algorithm almost loses all details of the sample. In order to mimic the real experimental situation, a 30 dB white noise is added into the simulation, the details of weakly scattering sample phase information can also be retrieved clearly by using the bright field of the diffraction patterns with high order iteration method. With the newly designed enhanced phase retrieval method for weakly scattering samples with partially coherent illumination, sample retrieval via PIE can not only use ordinary X-ray source or electron beam as illumination source, thereby avoiding the dependence on complete coherent source, but also obviously improve the retrieval quality of the sample characteristics, which widely expands the application fields of the PIE.

New Fourth Order Iterative Methods Second Derivative Free

In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives; therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

Ball Convergence for Second Derivative Free Methods in Banach Space

Hueso et al. (Appl Math Comput 211:190–197, 2009) considered a third and fourth order iterative methods for nonlinear systems. The methods were shown to of order third and fourth if the operator equation is defined on the j-dimensional Euclidean space (Hueso et al. in Appl Math Comput 211:190–197, 2009). The order of convergence was shown using hypotheses up to the third Frechet derivative of the operator involved although only the first derivative appears in these methods. In the present study we only use hypotheses on the first Frechet-derivative . This way the applicability of these methods is expanded. Moreover we present a radius of convergence a uniqueness result and computable error bounds based on Lipschitz constants. Numerical examples are also presented in this study.

Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions

We present a local convergence analysis of a sixth order iterative method for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Sharma et al. (Appl Math Comput 190:111---115, 2007) have shown convergence of these methods under hypotheses up to the fifth derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.

An Optimal Fourth Order Iterative Method for Solving Nonlinear Equations and Its Dynamics

We present a new fourth order method for finding simple roots of a nonlinear equation f(x)=0. In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.

A New Three Step Iterative Method without Second Derivative for Solving Nonlinear Equations

In this paper , an efficient new procedure is proposed to modify third –order iterative method obtained by Rostom and Fuad [Saeed. R. K. and Khthr. F.W. New third –order iterative method for solving nonlinear equations. J. Appl. Sci .7(2011): 916-921] , using three steps based on Newton equation , finite difference method and linear interpolation. Analysis of convergence is given to show the efficiency and the performance of the new method for solving nonlinear equations. The efficiency of the new method is demonstrated by numerical examples.

A New Three Step Iterative Method without Second Derivative for Solving Nonlinear Equations

In this paper , an efficient new procedure is proposed to modify third –order iterative method obtained by Rostom and Fuad [Saeed. R. K. and Khthr. F.W. New third –order iterative method for solving nonlinear equations. J. Appl. Sci .7(2011): 916-921] , using three steps based on Newton equation , finite difference method and linear interpolation. Analysis of convergence is given to show the efficiency and the performance of the new method for solving nonlinear equations. The efficiency of the new method is demonstrated by numerical examples.

A family of higher order multi-point iterative methods based on power mean for solving nonlinear equations

In this paper, we have presented a family of two-point fourth order, three-point sixth order and four-point twelfth order iterative methods without memory based on power mean using weight function. The family of fourth order methods is optimal in the sense of Kung-Traub hypothesis. In terms of computational point of view, our methods require three evaluations (one function and two first derivatives) to get fourth order, four evaluations (two functions and two derivatives) to get sixth order and five evaluations (three functions and two derivatives) to get twelfth order. Hence, these methods have high efficiency indices 1.587, 1.565 and 1.644 respectively. Few known results can be regarded as particular cases of our family of methods. Some numerical examples are tested to know the efficiencies of the methods which verify the theoretical results.

The Nonlinear Multi-Fuel Combustion Equation: Comparison between the Model of Heat in SIE for Ethanol-Gasoline Blended and High Order Iteration Method

A quasi one-dimensional engine cycle, the simulation program was developed to predict the simple heat transfer during combustion stroke in a spark ignition engine (SIE) of ethanol-gasoline blended fuels. A two-zone heat release model was utilized to model the combustion process inside the combustion cylinder by using high order iteration method (HIM) in combustion equation. The fuel, air and burned gas properties throughout the engine cycle were calculated by using variable specific heats. The transient heat flux inside the combustion chamber due to the change in the cylinder gas temperature and pressure during combustion was determined by using the constant of heat rate from enthalpy heat rate coefficient. The program was written in MATLAB. The results of this paper such as the average heat flux cycles of the burned gas during the combustion and showing the emissions of at during the combustion process by high iteration method. Based on these results, it was concluded that the fundamental thermodynamic properties of the fuel, air, and combustion product species were in agreement with previously published results to within the scatter of those results typically less than 1% form references 1%. The combustion products were calculated by the high order iteration methods by around 1% from the old methods.

Solving the one dimensional Bratu problem with efficient fourth order iterative methods

Efficient fourth order iterative methods are presented to approximate the solution of the one dimensional Bratu problem. These iterative methods improve the efficiency index of the Chebyshev and Newton methods. A semilocal convergence analysis assuming the same conditions for Newton’s method is performed. Finally, a particular case of the 1D Bratu problem is solved using directly these efficient methods.

Local Convergence for an Efficient Eighth Order Iterative Method with a Parameter for Solving Equations Under Weak Conditions

We present a local convergence analysis of an efficient eighth order iterative method with a parameter for approximate a locally unique solution of an equation defined on the real line. Earlier studies such as Bi et al. (J Comput Appl Math 225:105–112, 2009) have shown convergence of these methods under hypotheses up to the seventh derivative of the function although only the first derivative appears in the method. In this study we expand the applicability of these methods using only hypotheses up to the first derivative of the function. Numerical examples are also presented in this study.

Increasing the order of convergence for iterative methods to solve nonlinear systems

For solving nonlinear systems, we introduce a technique that improves the order of convergence of any given iterative method which uses the extended Newton iteration as a predictor. Based on a given iterative method of order $$p\ge 2$$pź2, a new method of order $$p+2$$p+2 is developed by introducing just one evaluation of the function. We obtain some new methods with higher order of convergence by applying this procedure to some known methods. Computational efficiency in the general form is discussed and comparisons are made between these new methods and the ones from which have been derived. We also perform several numerical tests to show the asymptotic behaviors which confirm the theoretical results.

Root finding by high order iterative methods based on quadratures

We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton–Cotes closed quadrature rules. We prove that when a quadrature rule with n + 1 nodes is used the resulting iterative method has convergence order at least n + 2, starting with the case n = 0 (which corresponds to the Newton’s method).

On the Nonlinear PerturbationK(n,m)Rosenau-Hyman Equation: A Model of Nonlinear Scattering Wave

We investigate a nonlinear wave phenomenon described by the perturbationK(n,m)Rosenau-Hyman equation within the concept of derivative with fractional order. We used the Caputo fractional derivative and we proposed an iteration method in order to find a particular solution of the extended perturbation equation. We proved the stability and the convergence of the suggested method for solving the extended equation without any restriction on(m,n)and also on the perturbations terms. Using the inner product we proved the uniqueness of the special solution. By choosing randomly the fractional orders andm, we presented the numerical solutions.

New Fourth and Fifth-Order Iterative Methods for Solving Nonlinear Equations

In this paper, we establish two new iterative methods of order four and five by using modified homotopy perturbation technique. We also present the convergence analysis of these iterative methods. To assess the validity and performance of these iterative methods, we have applied to solve some nonlinear problems.

New Modification of Fixed Point Iterative Method for Solving Nonlinear Equations

In this paper, we have modified fixed point method and have established two new iterative methods of order two and three. We have discussed their convergence analysis and comparison with some other existing iterative methods for solving nonlinear equations.

English

We establish here new three and four-step iterative methods of convergence order seven and fourteen to find roots of non-linear equations. The new developed iterative methods are variants of Newton's method that use different approximations of first derivatives in terms of previously known function values, thus improving efficiency indices of the methods. The seventh and fourteenth order iterative methods use four and five function values including one derivative of a function. So, the efficiency indices of these methods are , respectively. Numerical examples are given to show the performance of described methods.   Key words:  Non-linear equation, Iterative methods, convergence order, efficiency index.

A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations

We present a seventh order iterative method for solving systems of nonlinear equations. The algorithm is derivative free per full iteration. A comparison between the computational efficiencies of proposed technique and existing techniques of similar nature is made. It is shown that for large system the new method is more efficient. The performance is tested through numerical experimentation which also confirms the theoretical results.

On a two-parameter Chebyshev-Halley-like family of optimal two-point fourth order methods free from second derivatives

The one-parameter Chebyshev-Halley family is an important family of one-point third order iterative methods which requires one function, one first and one second derivative evaluations. The famous Chebyshev, Halley and Super-Halley methods are its members. Nedzhibov-Hasanov-Petkov (Numer. Alg. 42:127–136, 2006) approximated the second derivative present in the Chebyshev-Halley family to obtain a two-parameter Chebyshev-Halley-like family of two-point iterative methods free from second derivative. Only one member of this family known as the famous Jarratt method is fourth order and satisfies the Kung-Traub Conjecture. Other members are third order. Recent advancement in this field of numerical analysis have made it possible to develop fourth order methods from third order ones with the same number of function evaluations using weight functions. In this work, we develop a two-parameter Chebyshev-Halley-like family of two-point fourth order methods using weight functions. We compare the special members of the new family with the old one through an numerical example to illustrate the efficiency of the new family.